Question

If the equation 2ksinx + 7 = 4k – cos2x possesses a
solution for K E [a, b], then (a + b) is equal to​

Answer 1

Given : equation 2ksinx + 7 = 4k – cos2x possesses a  solution for k ∈ [a , b]

To Find : (a + b)  value

Solution:

2ksinx  + 7 = 4k - cos2x

=> 2ksinx  + 7 = 4k - (1- 2sin²x)

=> 2ksinx + 7 = 4k - 1 + 2sin²x

=> 2sin²x  - 2ksinx  + 4k  - 8  = 0

=>  sin²x  -  ksinx  + 2k  - 4 = 0

Sinx =  (k  ± √(k² - (4(2k-4)) ) /2

=> Sinx =  (k  ± √(k² - 8k + 16  ) /2

=> Sinx =  (k  ± √(k - 4)²  ) /2

=> Sinx =  (k + k - 4) /2  , ( k - k + 4)/2

=> sinx = (2k - 4)/2 ,        2  ( not possible)

=> sinx =  k - 2

sinx  range = - 1  to  1

=> k  =  1  to 3

k ∈ [1 , 3]

a + b = 1 + 3  =  4

(a + b) is equal to​ 4

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